Abstract
Abstract Fresán, Sabbah, and Yu constructed motives M n + 1 k ( Kl ) {\mathrm{M}_{n+1}^{k}(\mathrm{Kl})} over ℚ {\mathbb{Q}} encoding symmetric power moments of Kloosterman sums in n variables. When n = 1 {n=1} , they use the irregular Hodge filtration on the exponential mixed Hodge structure associated with M 2 k ( Kl ) {\mathrm{M}_{2}^{k}(\mathrm{Kl})} to compute the Hodge numbers of M 2 k ( Kl ) {\mathrm{M}_{2}^{k}(\mathrm{Kl})} , which turn out to be either 0 or 1. In this article, I explain how to compute the (irregular) Hodge numbers of M n + 1 k ( Kl ) {\mathrm{M}_{n+1}^{k}(\mathrm{Kl})} for n = 2 {n=2} or for general values of n such that gcd ( k , n + 1 ) = 1 {\gcd(k,n+1)=1} . I will also discuss related motives attached to Airy moments constructed by Sabbah and Yu. In particular, the computation shows that there are Hodge numbers bigger than 1 in most cases.
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